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A383729 Numbers k such that omega(k) = 5 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).

%I A383729 #10 Jun 09 2025 21:08:48
%S A383729 3570,7140,8970,10626,10710,14280,16530,17850,17940,20706,21252,21420,
%T A383729 24738,24882,24990,26910,28560,31878,32130,33060,35700,35880,36890,
%U A383729 38130,41412,42504,42840,44330,44850,49476,49590,49764,49938,49980,52170,53550,53820,54834,55986,57120
%N A383729 Numbers k such that omega(k) = 5 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).
%H A383729 Robert Israel, <a href="/A383729/b383729.txt">Table of n, a(n) for n = 1..10000</a>
%e A383729 10710 is a term because it has 5 distinct prime factors (2, 3, 5, 7 and 17) and the largest one is the sum of the others (2 + 3 + 5 + 7 = 17).
%p A383729 N:= 10^5: # for terms <= N
%p A383729 P:= select(isprime,[2,seq(i,i=3..N/(2*3*5*7),2)]):
%p A383729 V:= NULL:
%p A383729 i:= 1:
%p A383729 for j from i+1 while P[i]*P[j]^3*(P[i]+3*P[j]) < N do
%p A383729   for k from j+1 while P[i]*P[j]*P[k]^2*(P[i]+P[j]+2*P[k]) < N do
%p A383729     for l from k+1 while P[i]*P[j]*P[k]*P[l] * (P[i]+P[j]+P[k]+P[l]) <= N do
%p A383729       p5:= P[i]+P[j]+P[k]+P[l];
%p A383729       if not isprime(p5) then next fi;
%p A383729       for d1 from 1 while P[i]^d1 * P[j] * P[k] * P[l] * p5 <= N do
%p A383729        for d2 from 1 while P[i]^d1 * P[j]^d2 * P[k] * P[l] * p5 <= N do
%p A383729         for d3 from 1 while P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l] * p5 <= N do
%p A383729          for d4 from 1 while P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5 <= N do
%p A383729            for d5 from 1 while  P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5^d5 <= N do
%p A383729              V:= V,P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5^d5
%p A383729 od od od od od od od od:
%p A383729 sort([V]); # _Robert Israel_, Jun 09 2025
%t A383729 A383729Q[k_] := Length[#] == 5 && Total[Most[#]] == Last[#] & [FactorInteger[k][[All, 1]]];
%t A383729 Select[Range[10^5], A383729Q]
%Y A383729 Row n = 5 of A383726.
%Y A383729 Cf. A001221, A365795, A382469, A383725, A383726, A383728.
%K A383729 nonn
%O A383729 1,1
%A A383729 _Paolo Xausa_, May 08 2025